Complex Numbers and Quadratic Equations

The imaginary unit i is defined

The imaginary unit i is defined as √(-1), so i² = -1. A complex number is written as z = a + ib, where 'a' is the real part (Re z) and 'b' is the imag

Addition

Addition: (a + ib) + (c + id) = (a + c) + i(b + d). Subtraction: (a + ib) - (c + id) = (a - c) + i(b - d). Multiplication: (a + ib)(c + id) = (ac - bd

i¹ = i

i¹ = i, i² = -1, i³ = -i, i⁴ = 1. The pattern repeats every 4 powers. For any integer k: i⁴ᵏ = 1, i⁴ᵏ⁺¹ = i, i⁴ᵏ⁺² = -1, i⁴ᵏ⁺³ = -i. Example: i¹⁷ = i⁴

For z = a + ib

For z = a + ib: Modulus |z| = √(a² + b²) represents the distance from origin in Argand plane. Conjugate z̄ = a - ib is the reflection across real axis

For any positive real number

For any positive real number a: √(-a) = i√a. Example: √(-9) = 3i, √(-7) = i√7. Important: √(-a) × √(-b) ≠ √(ab) when both a, b > 0. Instead: √(-a) × √